Optimal. Leaf size=79 \[ \frac {2 \cos ^2(e+f x)^{5/4} (b \tan (e+f x))^{5/2} (a \sin (e+f x))^m \, _2F_1\left (\frac {5}{4},\frac {1}{4} (2 m+5);\frac {1}{4} (2 m+9);\sin ^2(e+f x)\right )}{b f (2 m+5)} \]
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Rubi [A] time = 0.12, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2602, 2577} \[ \frac {2 \cos ^2(e+f x)^{5/4} (b \tan (e+f x))^{5/2} (a \sin (e+f x))^m \, _2F_1\left (\frac {5}{4},\frac {1}{4} (2 m+5);\frac {1}{4} (2 m+9);\sin ^2(e+f x)\right )}{b f (2 m+5)} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2602
Rubi steps
\begin {align*} \int (a \sin (e+f x))^m (b \tan (e+f x))^{3/2} \, dx &=\frac {\left (a \cos ^{\frac {5}{2}}(e+f x) (b \tan (e+f x))^{5/2}\right ) \int \frac {(a \sin (e+f x))^{\frac {3}{2}+m}}{\cos ^{\frac {3}{2}}(e+f x)} \, dx}{b (a \sin (e+f x))^{5/2}}\\ &=\frac {2 \cos ^2(e+f x)^{5/4} \, _2F_1\left (\frac {5}{4},\frac {1}{4} (5+2 m);\frac {1}{4} (9+2 m);\sin ^2(e+f x)\right ) (a \sin (e+f x))^m (b \tan (e+f x))^{5/2}}{b f (5+2 m)}\\ \end {align*}
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Mathematica [A] time = 8.14, size = 87, normalized size = 1.10 \[ \frac {2 (b \tan (e+f x))^{5/2} \sec ^2(e+f x)^{m/2} (a \sin (e+f x))^m \, _2F_1\left (\frac {m+2}{2},\frac {1}{4} (2 m+5);\frac {1}{4} (2 m+9);-\tan ^2(e+f x)\right )}{b f (2 m+5)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \tan \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{m} b \tan \left (f x + e\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \sin \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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